[General] We agreed to do a skill exchange....

[Silverobama]

I met a woman in the gym. She swims very well. One day she heard that I spoke English and she asked if I could teach her English. I then said sure and I could learn how to swim from her. She said she'd love to teach me.

I wrote a sentence to express this idea:

We agreed to do a skill exchange. She teaches me how to swim while I teach her English.

Is my italic sentence natural? This is not a dialog.

 

[jutfrank]

I think it's fine. Pluralise to skills, though.

 

[Tarheel]

I met a woman in the gym. She swims very well. One day she heard that I speak English and she asked if I could teach her English. I then said "Sure!" and could I learn how to swim from her? She said she'd love to teach me.

I wrote a sentence to express this idea:

We agreed to do a skill exchange. She teaches me how to swim and I teach her English.

Is my italic sentence natural? This is not a dialog.

I wouldn't use "while" unless the two things were happening at the same time.

 

[Silverobama]

I think it's fine. Pluralise to skills, though.

We agreed to do skills exchange. She teaches me how to swim while I teach her English.

Is it okay now?

 

[Tarheel]

Yes, but since you haven't started yet I would probably put it in the future tense, thus:

She will teach me how to swim, and I will teach her how to speak English.

 

[GoesStation]

We agreed to do a skills exchange. She teaches me how to swim while I teach her English.

Is it okay now?

See above.